3.1.16 \(\int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx\) [16]

Optimal. Leaf size=85 \[ \frac {10 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{21 b}+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {10 \sin (a+b x)}{21 b \sqrt {\sec (a+b x)}} \]

[Out]

2/7*sin(b*x+a)/b/sec(b*x+a)^(5/2)+10/21*sin(b*x+a)/b/sec(b*x+a)^(1/2)+10/21*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1
/2*a+1/2*b*x)*EllipticF(sin(1/2*a+1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)*sec(b*x+a)^(1/2)/b

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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2720} \begin {gather*} \frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {10 \sin (a+b x)}{21 b \sqrt {\sec (a+b x)}}+\frac {10 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{21 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^(-7/2),x]

[Out]

(10*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(21*b) + (2*Sin[a + b*x])/(7*b*Sec[a + b*
x]^(5/2)) + (10*Sin[a + b*x])/(21*b*Sqrt[Sec[a + b*x]])

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

\begin {align*} \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx &=\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \int \frac {1}{\sec ^{\frac {3}{2}}(a+b x)} \, dx\\ &=\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {10 \sin (a+b x)}{21 b \sqrt {\sec (a+b x)}}+\frac {5}{21} \int \sqrt {\sec (a+b x)} \, dx\\ &=\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {10 \sin (a+b x)}{21 b \sqrt {\sec (a+b x)}}+\frac {1}{21} \left (5 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx\\ &=\frac {10 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{21 b}+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {10 \sin (a+b x)}{21 b \sqrt {\sec (a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 61, normalized size = 0.72 \begin {gather*} \frac {\sqrt {\sec (a+b x)} \left (40 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+26 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{84 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^(-7/2),x]

[Out]

(Sqrt[Sec[a + b*x]]*(40*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 26*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)
]))/(84*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(97)=194\).
time = 2.58, size = 199, normalized size = 2.34

method	result	size
default	\(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (48 \left (\cos ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{21 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sec(b*x+a)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/21*((2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*(48*cos(1/2*b*x+1/2*a)^9-120*cos(1/2*b*x+1/2*a)^
7+128*cos(1/2*b*x+1/2*a)^5-72*cos(1/2*b*x+1/2*a)^3+5*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cos(1/2*b*x+1/2*a)^2+1)^
(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))+16*cos(1/2*b*x+1/2*a))/(-2*sin(1/2*b*x+1/2*a)^4+sin(1/2*b*x+1/2*a)
^2)^(1/2)/sin(1/2*b*x+1/2*a)/(2*cos(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(sec(b*x + a)^(-7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.69, size = 87, normalized size = 1.02 \begin {gather*} \frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} + 5 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{\sqrt {\cos \left (b x + a\right )}} - 5 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

1/21*(2*(3*cos(b*x + a)^3 + 5*cos(b*x + a))*sin(b*x + a)/sqrt(cos(b*x + a)) - 5*I*sqrt(2)*weierstrassPInverse(
-4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*I*sqrt(2)*weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)))
/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sec ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(b*x+a)**(7/2),x)

[Out]

Integral(sec(a + b*x)**(-7/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(b*x+a)^(7/2),x, algorithm="giac")

[Out]

integrate(sec(b*x + a)^(-7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/cos(a + b*x))^(7/2),x)

[Out]

int(1/(1/cos(a + b*x))^(7/2), x)

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